Excess and saturated D-optimal designs for the rational model

Abstract

For a rational two-dimensional nonlinear in parameters model used in analytical chemistry, we investigate how homothetic transformations of the design space affect the number of support points in the optimal designs. We show that there exist two types of optimal designs: a saturated design (i.e. a design with the number of support points which is equal to the number of parameters) and an excess design (i.e. a design with the number of support points which is greater than the number of parameters). The optimal saturated designs are constructed explicitly. Numerical methods for constructing optimal excess designs are used.

Appendix

Throughout this section without losing generality, we can assume that $\gamma =1$. Also, due to the fact that the D-optimal design for model (3.1) doesn’t depend on the parameter $\theta _0$, we put $\theta _0\equiv 1.$ To prove Theorem 2 we need to establish the following lemma first:

Lemma 1

A saturated D-optimal design for model (3.1) is concentrated at points

$$\begin{aligned} \{A=( b_1/(2+b_1\theta _1),0), \ B=(b_1, 0), \ C=(\alpha _1, \alpha _2) \}, \ \alpha _i \in (0,1] \end{aligned}$$

(6.1)

where the point C coincides with one of three points: $(b_1, (1+b_1\theta _1)/\theta _2)$, $(b_1, b_2)$, $((1+b_2\theta _2)/\theta _1, b_2)$.

Proof of Lemma 1

According to Theorem 1 a design $\xi$ is D-optimal for model (3.1) if it satisfies the equation

$$\begin{aligned} \max _{(x_1,x_2)\in {\mathcal {X}}}d((x_1,x_2),\xi )=3,&\\ \hbox {where}\ d((x_1,x_2),\xi )=&f^T((x_1,x_2))M^{-1}(\xi )f((x_1,x_2)). \end{aligned}$$

For model (3.1) the function $d((x_1,x_2),\xi )$ has a form:

$$\begin{aligned} d((x_1,x_2),\xi )=\frac{x_1^2(x_1^2a_1+x_1x_2a_2+x_2^2a_3+x_1a_4 +x_2a_5+a_6)}{(x_1\theta _1+x_2\theta _2+1)^4}, \end{aligned}$$

where $a_i$ are coefficients depending on $\theta _i$ and elements of the matrix $M^{-1}(\xi )$. The analysis of this function shows that it has no more than 4 maxima at points

$$\begin{aligned}&\{\breve{A}=( \alpha _1,0), \ B=(b_1, 0), \ \breve{C}=(b_1, \beta _1),\ \breve{D}=(\alpha _2, \beta _2 ) \},\nonumber \\&\quad \alpha _1,\alpha _2\in (0,b_1], \ \beta _1,\beta _2\in (0,b_2]. \end{aligned}$$

(6.2)

Indeed, for any given non-singular design $\xi$ and fixed $x_1=x_1^{*}\in (0,b_1]$ the function $d((x_1^{*},x_2),\xi )$ has no more than two maxima at the point $x_2=0$ and at point $x_2=x_2^{*}\in (0,b_2]$ on the interval $[0,b_2]$. On the other hand, for fixed $x_2=x_2^{*}\in [0,b_2]$ the function $d((x_1,x_2^{*}),\xi )$ has also no more than two maxima at the point $x_1=b_1$ and at the point $x_1=x_1^{*}\in (0,b_1)$ on the interval $[0,b_1].$ This immediately implies that the function $d((x_1,x_2),\xi )$ has maxima at points of the form (6.2). We have to show now that the number of global maxima is less than or equal to 4. To do this, we prove that the function $d((x_1,x_2),\xi )$ cannot have two global maxima at points of the form $\breve{D}$ when $\alpha _2<b_1,\ \beta _2<b_2$. We prove it by reductio ad absurdum. Let

$$\begin{aligned} \max _{x_1,x_2}d((x_1,x_2),\ \xi )=d(X^{*}_1,\ \xi )=d(X^{*}_2,\ \xi ),\ \\ X^{*}_i=(x^{*}_{i1},x^{*}_{i2})\in [0,b_1)\times [0,b_2),\ i=1,2. \end{aligned}$$

Consider the line $x_2=ax_1+b$ that passes through the points $X^{*}_i$, $i=1,2.$ We have

$$\begin{aligned} \max _{x_1}d((x_1,ax_1+b),\ \xi )=d((x^{*}_{11},ax^{*}_{11}+b),\ \xi )=d((x^{*}_{21},ax^{*}_{21}+b),\ \xi ). \end{aligned}$$

After the appropriate replacements, we obtain

$$\begin{aligned} d((x_1,ax_1+b),\xi )= \frac{x_1^2({\widetilde{a}}x_1^2+{\widetilde{b}}x_1+{\widetilde{c}}_1)}{(x_1+ {\widetilde{c}}_2)^4}. \end{aligned}$$

(6.3)

Note that the function $(x_1+{\widetilde{c}}_2)^{-4}$ is bounded and strictly increasing (or decreasing) on the interval $[0,b_1]$. On the other hand, the function $x_1^2({\widetilde{a}}x_1^2+{\widetilde{b}}x_1+{\widetilde{c}})$ has no more than 2 global maxima on the interval $[0,b_1]$ and one of them has to be located at the point $x_1=b_1$. It follows from this that the function in (6.3) has no more than 2 global maxima at the points $x_1=b_1$ and $x_1=x^{*}_1\in (0,b_1)$ on the interval $[0,b_1]$. Thus, we have obtained a contradiction. Therefore, we have proved that the saturated D-optimal design is concentrated at points

$$\begin{aligned} \text{ supp }(\xi )= & {} \{\breve{A}=(\alpha _1, 0), \ B=(b_1,0),\ \breve{D}=(\alpha _2, \beta _1)\}, \\&\alpha _1 \in (0,b_1),\ \alpha _2 \in (0,b_1],\ \beta _1 \in (0,b_2]. \end{aligned}$$

Now we have to consider all possible combinations of sets of parameter’s values $\{\alpha _1,\alpha _2,\beta _1\}$ to show that any saturated D-optimal design is concentrated at points (6.1). Since $0<\alpha _1<b_1$ we have only 4 combinations: $\{0<\alpha _1<b_1, \alpha _2=b_1, 0<\beta _1<b_2\}$, $\{0<\alpha _1<b_1, \alpha _2=b_1, \beta _1=b_2\}$, $\{0<\alpha _1<b_1, 0<\alpha _2<b_1, \beta _1=b_2\}$, $\{0<\alpha _1<b_1, 0<\alpha _2<b_1, 0<\beta _1<b_2\}$. By Theorem 1 any saturated D-optimal design $\xi ^*$ must satisfy the conditions:

$$\begin{aligned} \max _{(x_1,x_2)\in {\mathcal {X}}}d((x_1,x_2),\xi ^*)=d((t_{i1},t_{i2}),\xi ^*)=3,\ \forall \ (t_{i1},t_{i2}) \in \text{ supp }(\xi ^*). \end{aligned}$$

This implies that if $(t_{i1},t_{i2})$ is an inner point of the design space ${\mathcal {X}}$ then it satisfies the system of equations:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial d((x_1,x_2),\xi ^*)}{\partial x_1}\Bigr |_{x_1=t_{i1},x_2=t_{i2}}=0,\\ \displaystyle \frac{\partial d((x_1,x_2),\xi ^*)}{\partial x_2}\Bigr |_{x_1=t_{i1}, x_2=t_{i2}}=0.\\ \end{array} \right. \end{aligned}$$

Thus, for our four combinations we have four different systems of corresponding equations. Since

$$\begin{aligned} \frac{\partial d((x_1,x_2),{\overline{\xi }})}{\partial x_1}\Bigr |_{x_1=\alpha _1,x_2=0}=0 \Rightarrow -b_1+(\theta _1b_1+2)\alpha _1=0 \Rightarrow \alpha _1=\frac{b_1}{\theta _1b_1+2} \end{aligned}$$

we immediately obtain that any saturated D-optimal design is concentrated at points (6.1). Note that first 3 combinations give the support points of designs (3.4), (3.5), (3.6) and for the last one ($\{0<\alpha _2<1, 0<\beta _1<1\}$) there is no solution:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial d((x_1,x_2),\xi ^*)}{\partial x_1}\Bigr |_{x_1=\alpha _1,x_2=0}=0\\ \displaystyle \frac{\partial d((x_1,x_2),\xi ^*)}{\partial x_1}\Bigr |_{x_1=\alpha _2, x_2=\beta _1}=0\\ \displaystyle \frac{\partial d((x_1,x_2),\xi ^*)}{\partial x_2}\Bigr |_{x_1=\alpha _2, x_2=\beta _1}=0\\ \end{array} \right. \Rightarrow \left\{ \begin{array}{l} \displaystyle -b_1+(\theta _1b_1+2)\alpha _1=0 \\ \displaystyle \alpha _2\theta _1-\beta _1\theta _2+1=0\\ \displaystyle \alpha _2\theta _1-\beta _1\theta _2-1=0\\ \end{array} \right. \end{aligned}$$

Lemma 1 is proved. $\square$

Proof of Theorem 2

Without losing generality, we can assume that $b_1=b_2=1$.

Fig. 9

The behavior of the function $d((t_1,t_2),{\bar{\xi }}_1^*)$ when $\theta _1=1,\theta _2=3$, ${\mathcal {X}}=[0,1]\times [0,1]$ for the case (3.4) from Theorem 2

Optimality of designs (3.4)–(3.6) is verified directly by Theorem 1. For example, in case (a) for the design ${\bar{\xi }}_1^*$ in form (3.4) (under our assumption) we have

$$\begin{aligned} {\bar{\xi }}_1^*= & {} \left( \begin{array}{ccc} (1/(2+\theta _1),0)&{} (1, 0)&{} (1, (1+\theta _1)/\theta _2)\\ 1/3&{}1/3&{}1/3 \end{array}\right) , \ \theta _2\ge \theta _1+1, \\ d((t_1,t_2),{\bar{\xi }}_1^*)= & {} \frac{3\left( 1+\theta _1\right) ^2 t_1^2}{(1+\theta _1t_1+\theta _2t_2)^4}\sum _{0\le i+j\le 2}a_{ij}t_1^it_2^j, \\ a_{20}= & {} \theta _1^2+4\theta _1+20, \quad a_{11} = -2\theta _1\theta _2-4\theta _2, \quad a_{02}=17\theta _2^2, \\ a_{10}= & {} -2\theta _1-36, \quad a_{01} = 2\theta _2, \\ a_{00}= & {} 17. \end{aligned}$$

There are two stationary points of the function $d((t_1,t_2),{\bar{\xi }}_1^*)$ on $[0,1]\times [0,1]:$

$$\begin{aligned} Q_1= & {} \Bigl (\frac{2}{\theta _1+3}, \frac{\theta _1+1}{\theta _2(\theta _1+3)} \Bigr ), \quad Q_2=\Bigl (\frac{6}{\theta _1+7}, \frac{\theta _1+1}{\theta _2(\theta _1+7)} \Bigr ). \end{aligned}$$

(6.4)

The function $d((t_1,t_2),{\bar{\xi }}_1^*)$ has the following values at these points:

$$\begin{aligned} d(Q_1,{\bar{\xi }}_1^*)=\frac{3}{2}, \quad d(Q_2,{\bar{\xi }}_1^*)=\frac{81}{64}. \end{aligned}$$

(6.5)

The Hessian matrix of this function is

$$\begin{aligned} H =\frac{\partial ^2}{\partial t_i\partial t_j}d((t_1,t_2),{\bar{\xi }}_1^*)\in R^{2\times 2}. \end{aligned}$$

Direct calculations show that for eigenvalues $\mu _{1,2}(Q_1)$ of the matrix H at the point $Q_1$ the inequalities $\mu _1(Q_1)>0$ and $\mu _2(Q_2)<0$ hold. Thus, $Q_1$ is a saddle point. On the other hand, for the point $Q_2$ the inequalities $\mu _1(Q_1)>0$ and $\mu _2(Q_2)>0$ hold and this entails that $Q_2$ is a minimum point.

Since there are no stationary points $Q_i$ of the function $d((t_1,t_2),{\bar{\xi }}_1^*)$ on $[0,1]\times [0,1]$ such that $d(Q_i, ,{\bar{\xi }}_1^*)\le 3$ this function reaches its maxima at boundary points. Since $d((0,t_2),{\bar{\xi }}_1^*)\equiv 0$ we have only four possible candidates:

$$\begin{aligned} \{\breve{A}=( \alpha _1,0), \ B=(1, 0), \ \breve{C}=(1, \beta _1),\ D=(1, 1 ) \}, \ \alpha _1,\beta _1 \in (0,1). \end{aligned}$$

For the design ${\bar{\xi }}_1^*$ with $\breve{A}=(( 1/(2+\theta _1),0),\ \breve{C}=(1, (1+\theta _1)/\theta _2))$ we have

$$\begin{aligned} d(\breve{A},{\bar{\xi }}_1^*)=d(B,{\bar{\xi }}_1^*)=d(\breve{C},{\bar{\xi }}_1^*)=3 \end{aligned}$$

and

$$\begin{aligned} d(D,{\bar{\xi }}_1^*)=\frac{3(\theta _1+1)^2(\theta _1^2+(-2\theta _2+2) \theta _1+17\theta _2^2-2\theta _2+1)}{(\theta _1+\theta _2+1)^4}. \end{aligned}$$

Note that under fixed $\theta _1$ this function decreases by $\theta _2$ on the interval $[\theta _1+1,\infty ).$ Indeed,

$$\begin{aligned} \frac{\partial d(D,{\bar{\xi }}_1^*)}{\partial \theta _2}=-\frac{6(3\theta _1-17\theta _2+3)(\theta _1+1)^2(\theta _1-\theta _2+1)}{(\theta _1+\theta _2+1)^5} \le 0,\ \ \theta _2\in [\theta _1+1,\infty ). \end{aligned}$$

Since for case (a) the inequality $\theta _2\ge \theta _1+1$ holds, we have:

$$\begin{aligned} d(D,{\bar{\xi }}_1^*)\le d(D,{\bar{\xi }}_1^*)\Bigr |_{\theta _2=\theta _1+1}=3. \end{aligned}$$

Thus, by Theorem 1 the design in (3.4) for case (a) is D-optimal. The remaining cases are verified in a similar way.

The behavior of the function $d((t_1,t_2),{\bar{\xi }}_1^*)$ for $\theta _1=1,\theta _2=3$, ${\mathcal {X}}=[0,1]\times [0,1]$ is depicted in Fig. 9.

Theorem 2 is proved. $\square$